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López Jorge M. - Elementary analysis: the theory of calculus

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López Jorge M. Elementary analysis: the theory of calculus
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Kenneth A. Ross Undergraduate Texts in Mathematics Elementary Analysis 2nd ed. 2013 The Theory of Calculus 10.1007/978-1-4614-6271-2_1 Springer Science+Business Media New York 2013
1. Introduction
Kenneth A. Ross 1
(1)
Department of Mathematics, University of Oregon, Eugene, OR, USA
Abstract
The underlying space for all the analysis in this book is the set of real numbers. In this chapter we set down some basic properties of this set. These properties will serve as our axioms in the sense that it is possible to derive all the properties of the real numbers using only these axioms. However, we will avoid getting bogged down in this endeavor. Some readers may wish to refer to the appendix on set notation.
The underlying space for all the analysis in this book is the set of real numbers. In this chapter we set down some basic properties of this set. These properties will serve as our axioms in the sense that it is possible to derive all the properties of the real numbers using only these axioms. However, we will avoid getting bogged down in this endeavor. Some readers may wish to refer to the appendix on set notation.
1 The Set Picture 1 of Natural Numbers
We denote the set {1,2,3,} of all positive integers by Picture 2 . Each positive integer n has a successor, namely n +1. Thus the successor of 2 is 3, and 37 is the successor of 36. You will probably agree that the following properties of Picture 3 are obvious; at least the first four are.
N1.
1 belongs to Picture 4 .
N2.
If n belongs to Picture 5 , then its successor n +1 belongs to Picture 6 .
N3.
1 is not the successor of any element in Picture 7 .
N4.
If n and m in Picture 8 have the same successor, then n = m .
N5.
A subset of Picture 9 which contains 1, and which contains n +1 whenever it contains n , must equal Picture 10 .
Properties N1 through N5 are known as the Peano Axioms or Peano Postulates . It turns out most familiar properties of Picture 11 can be proved based on these five axioms; see [].
Lets focus our attention on axiom N5, the one axiom that may not be obvious. Here is what the axiom is saying. Consider a subset S of Picture 12 as described in N5. Then 1 belongs to S . Since S contains n +1 whenever it contains n , it follows that S contains Picture 13 . Again, since S contains n +1 whenever it contains n , it follows that S contains Picture 14 . Once again, since S contains n +1 whenever it contains n , it follows that S contains Picture 15 . We could continue this monotonous line of reasoning to conclude S contains any number in Picture 16 . Thus it seems reasonable to conclude Picture 17 . It is this reasonable conclusion that is asserted by axiom N5.
Here is another way to view axiom N5. Assume axiom N5 is false. Then Picture 18 contains a set S such that
(i)
1 S ,
(ii)
If n S , then n +1 S ,
and yet Elementary analysis the theory of calculus - image 19 . Consider the smallest member of the set Elementary analysis the theory of calculus - image 20 , call it n 0. Since (i) holds, it is clear n 01. So n 0 is a successor to some number in Elementary analysis the theory of calculus - image 21 , namely n 01. We have n 01 S since n 0 is the smallest member of Elementary analysis the theory of calculus - image 22 . By (ii), the successor of n 01, namely n 0, is also in S , which is a contradiction. This discussion may be plausible, but we emphasize that we have not proved axiom N5 using the successor notion and axioms N1 through N4, because we implicitly used two unproven facts. We assumed every nonempty subset of Picture 23 contains a least element and we assumed that if n 01 then n 0 is the successor to some number in Elementary analysis the theory of calculus - image 24 .
Axiom N5 is the basis of mathematical induction. Let Elementary analysis the theory of calculus - image 25 be a list of statements or propositions that may or may not be true. The principle of mathematical induction asserts all the statements Elementary analysis the theory of calculus - image 26 are true provided
(I1 ) P 1 is true,
(I2 ) P n +1 is true whenever P n is true.
We will refer to (I1), i.e., the fact that P 1 is true, as the basis for induction and we will refer to (I2) as the induction step. For a sound proof based on mathematical induction, properties (I1) and (I2) must both be verified. In practice, (I1) will be easy to check.
Example 1
Prove Elementary analysis the theory of calculus - image 27 for positive integers n .
Solution
Our n th proposition is
Elementary analysis the theory of calculus - image 28
Thus P 1 asserts Elementary analysis the theory of calculus - image 29 , P 2 asserts Elementary analysis the theory of calculus - image 30 , P 37 asserts etc In particular P 1 is a true assertion which serves as our basis for - photo 31
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